Double-elliptic integrable systems

The talk concerns the study of the p,q-duality in the theory of integrable systems. In particular, we will focus on the recent works, where a new approach to obtain the multi-particle generalization of the elliptic p–q dual integrable systems was developed. The list of these systems includes duals to elliptic Calogero and elliptic Ruijsenaars models. The most interesting are the double-elliptic integrable systems, where both coordinates and momenta take values on elliptic curves. From the point of view of the low-energy effective actions of Yang–Mills theories these integrable systems are associated with (compactified) six-dimensional SUSY gauge theories and are strongly involved in modern discussion of the Seiberg–Witten theory.